From Tolerance Near Sets to Perceptual Image Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 178

Definition 9.4.1 Nearness Relation
LethO, F ibe perceptual system and let X, Y⊆O. A set X is near to a set Y within

the perceptual systemhO, and there are

A∈O /∼ A , B∈O /∼ B , C∈O /∼ f such that A, B⊆C. If a perceptual system is understood,
then we say briefly that a set X is near to a set Y .

F i(X ./ F Y ) iff there areA,B⊆ F and f∈ F

Definition 9.4.2 Weak Nearness Relation
LethO,

ibe a perceptual system and let X, Y⊆O. A set X is weakly near to a set Y
within the perceptual systemhO,

F

F i(X./ F Y ) iff there are x∈X and y∈Y and there is

B⊆ F such that x∼ B y, i.e., set X is weakly near to a set Y .

TABLE 9.1 Relation Symbols
Symbol Interpretation
O Set of perceptual objects,
X X⊆O, set of sample objects,
F A set of probe functions,
B B⊆F,
< reals,
φ φ∈B, where φ i : O−→<, probe function,
∼ B {(x, y)| f (x) = f (y)∀φ∈B},
' B {(x, y)∈O×O|∃φ i ∈B, φ i (x) = φ i (y)},
∼ = B,ε Perceptual tolerance relation,
x /∼ B x /∼ B ={y∈X | y∼ B x}(class),
O /∼ B O /∼ B ={x /∼ B | x∈O}, quotient set,
./ B nearness relation symbol,
./ B weak nearness relation symbol,
./ B,ε

Weak tolerance nearness relation symbol.

Definition 9.4.3 Weak Tolerance Nearness

Let O,

be a perceptual system and let X, Y⊆O, ∈

F R . The set X is perceptually near

to the set Y within the perceptual system O,

(X ./ F

F Y ) iff there exists x∈X, y∈Y

such that x ∼ = B y. If a perceptual system is understood, then
we say shortly that a set X is perceptually near to a set Y in a weak tolerance sense of
nearness.

and there is a φ∈ F , ∈ R

Definition 9.4.4 Tolerance Near Sets
LethO, ibe a perceptual system and let X⊆O. A set X is a tolerance near set iff there is

Y⊆O such that X ./ F

F

Y . The family of near sets of a perceptual systemhO, F iis denoted

by Near F (O).

9.5 Three Tolerance Near Set-based Nearness Measures for

Image Analysis and Comparison

The term tolerance space was introduced by E.C. Zeeman in 1961 in modelling visual per-
ception with tolerances. Tolerance relations are viewed as good models of how one perceives,
how one sees (Zeeman, 1961). Tolerance relations are also considered as a basis for studying
similarities between objects (Peters, 2008c). The basic idea behind using tolerance classes